On Multicomponent System Reliability with Microshocks - Microdamages Type of Components Interaction

Transcription

1 On Mulicomponen Sysem Reliabiliy wih Microshocks - Microdamages Type of Componens Ineracion Jerzy K. Filus, and Lidia Z. Filus Absrac Consider a wo componen parallel sysem. The defined new sochasic dependences of he componen life-imes are undersood as a sochasic reflecion of raher complicaed componens physical ineracions. The ineracions are assumed o obey some coninuous paern, valid unil one of he componens fails. According o ha paern some infiniesimal micro-damages in each componen physical srucure (caused by he remaining componen s harmful aciviy) are relaed o corresponding infiniesimal incremens in he componen failure rae. Based on his associaion beween he physical failure mechanism and is much simpler probabilisic equivalence, we describe he laer analyically in erms of he lifeimes join probabiliy disribuion; acually we consruced a wide class of such sochasic models. As a special case of he bivariae disribuions we obain he firs bivariae Gumbel disribuion along wih is, perhaps firs known, reliabiliy inerpreaion. Inde Terms Gumbel s bivariae eponenial disribuion as a reliabiliy model, Sochasic dependence, Sysem reliabiliy models. I. INTRODUCTION We consruc a (new) sochasic model for reliabiliy ([]) of a wo componens parallel J. K. Filus is wih he Deparmen of Mahemaics and Compuer Science, Oakon Communiy College, Des Plaines, IL 6006, USA ( jfilus@oakon.edu) L. Z. Filus is wih he Deparmen of Mahemaics, Norheasern Illinois Universiy, Chicago, IL 6065, USA ( L-Filus@neiu.edu) sysem. Boh he componens u, u, also called unis, are considered in wo differen siuaions ha we call he offsysem and he in-sysem condiions. While in he off -sysem condiions boh he unis work separaely of each oher, i. e., wih no muual physical ineracions. As a consequence of lack of physical conac, heir life-imes, say T, T are sochasically independen random variables wih he, given in advance, original probabiliy disribuion funcions F( ), F( ). The insysem condiions are creaed when he componens are insalled ino he sysem. I is assumed, ha in he sysem some muual ineracions occur. As a resul, in he considered sysem, side-effecs of each componen performance, influence physics of failure mechanism of he remaining uni. The componens failure mechanism has wo parallel aspecs: ) he realiy aspec, based on engineering devices and / or underlying physical and chemical processes ha evenually lead o each uni s failure, and ) he sochasic reflecion of he sysem s physics. The associaed physical phenomena are ofen oo comple and complicaed o be followed and efficienly handled for he reliabiliy predicion purposes. On he physical par of his (modeling) problem, he muual impac of any componen on he oher, can be eplained in he following manner. During he componens in-sysem

2 performance, eiher of he wo creaes such an (environmenal) siuaion ha he oher uni is consanly bombarded by a sring of small harmful (or beneficial) microincenives ha we call, in his paper, micro-shocks. Each such micro-shock causes a corresponding micro-damage in he affeced uni s physical consiuion. This micro-shock micro-damage relaion we classify as being he physics of he problem. This par migh be invesigaed by use of engineering mehods as long as he underlying phenomena are no oo complicaed for an efficien physical analysis. Oherwise, in order o simplify invesigaions, we raher consider an associaed sochasic mechanism, basically ignoring he deails of underlying physical phenomena and replacing he problem s physics by proper saisical analysis of he sochasic model now being consruced. For he juncion beween he sysem s physics and is probabilisic behavior we will consider any possible relaion beween he micro-damages of he componens and he resuling small probabiliy -microchanges in he original probabiliy disribuions of he uni life-imes. This sochasic par of he problem we may consider as he micro-damages probabiliy-micro-changes relaion ha in accordance o he chosen mehodology of he research replaces he previous, more physical, relaion. In order o quanify hese micro-changes in he life ime probabiliy disribuions we have chosen, among several oher possibiliies, micro-changes (in general incremens) in he componens failure raes. Denoe he componens (original) offsysem failure raes, associaed wih he independen life imes T, T by he symbols λ ( ), λ ( ), respecively. I is quie obvious ha he changes in λ ( ), λ ( ) will ransform he independen offsysem life imes T, T ino some new dependen in sysem life imes X, X. Now we can formulae he paper s objecive as o deermine he join reliabiliy funcion of he lifeimes random vecor (X, X ). This funcion, equivalen of he join probabiliy disribuion, will serve as he model of he considered sysem reliabiliy. Oher, similar sochasic models one can find in quie a large subjec s lieraure. See, for eample, [3] [7]. To sar wih he model consrucion based on he micro-damages probabiliymicro-changes paern we mus o ake ino consideraion ha each micro-damage ha occurs is eremely small so ha here is usually no pracical possibiliy as o deec any significan influence of his microdamage on a given uni reliabiliy. This can only be realized afer a long enough ime period as he micro-damages cumulae heir effecs. Since also he ime periods he micro-damages occur are eremely small, while he number of hese micro-evens per ime uni is a huge one, we have chosen he (ypical in such siuaions) coninuous approimaion as an analyical descripion of hese fugiive phenomena. In oher words we uilize he familiar calculus noion of he infiniesimal quaniies. Thus, in he classical analyical model, an accumulaion of he microchanges in boh he componens failure raes is o be epressed by he Riemann inegral device. II. THE MODEL S CONSTRUCTION In his secion we find he join probabiliy disribuion of he random vecor (X,X ) in erms of is join survival funcion S(, ) = Pr(X >, X > ). Recall, ha he

3 join survival funcion s*(, ) of he independen off-sysem componen lifeimes T, T is given by he following produc form: s*(, ) = ep[- 0 λ ( ) d - 0 λ ( ) d ] () where λ ( ), λ ( ) are he componen s off-sysem failure raes. When he componens u, u work in he sysem hen, in accordance wih he adoped (in he analyical model) assumpion, in every infiniesimal small ime inerval [τ k, τ k + d τ k ) an occurrence of an infiniesimal micro-damage of he componen, say u k, k =, ; (ha is caused by side effecs accompanying an aciviy of componen u m, where m =, ; m k) resuls in an infiniesimal incremen α k m (τ k ) d τ k of u k s failure rae. For every pas ime insan τ k ha incremen is given by a predeermined quaniy α km (τ k ) which, in a sochasic way, reflecs an amoun (or a densiy ) of he physical influence of he componen u m on componen u k a a given ime insan τ k. Tha quaniy is chosen (and hen mus be saisically verified for fi o available daa) o be a coninuous funcion of all he pas epochs τ k he micro-damages occurred. In par i sands for ime derivaive of an overall failure rae. In he mahemaical model, all he sochasic effecs α k m (τ k )d τ k (of he physical micro-damage s ) sum up over he ime. Each of heir parial accumulaion (creaed from he ime zero, up o he epoch, say k considered o be curren ) is epressed as he following Riemann inegral: k ϕ k ( k ) = ( 0 α k m (τ k ) d τ k, ( k m ). () This inegral iself is a non decreasing coninuous funcion of he curren ime k as is aken over all ime inervals [ 0, k], wih 0 k k <, for k =,. Boh he variables k, ( k =, ) as presen in he foregoing condiion, are he same as he argumens of he survival funcion (). The inegral () will be hough of as a measure of magniude of he u k s microdamage s accumulaion up o he curren ime epoch k. A every curren ime insan k, he overall in-sysem failure rae r k ( k ) of componen u k ( k =, ) is defined o be a simple arihmeic sum of he off-sysem failure rae λ k ( k ) and he addiional failure rae given by he inegral (). Thus, a every ime epoch k one obains he following formula for he in-sysem failure rae r k ( k) of componen u k : k r k ( k ) = λ k ( k ) + α k m (τ k ) d τ k (3) 0 as k, m =,, and k m. We consider he failure rae formula (3) o be valid for each ime argumen k saisfying 0 k, where = minimum (, ) is considered o be he ime of he firs failure in he sysem. From he above one obains he following survival funcion: S () = Pr ( min(x, X ) > ), wih = min(, ) ), for he firs order saisics X of he se of random variables: { X, X }. Tha is: S () = ep[ - r ( ) d - r ( ) d ] 0 0 (4)

4 where r ( ) and r ( ) are given by () for k =,. These wo funcions represen he in sysem failure raes of he componens u and u respecively, a he ime insances,, boh prior o he ime, say, afer which he firs failure in he sysem occurs. Consequenly, he given by (4) funcion S () is he (whole) sysem reliabiliy funcion, if he sysem reliabiliy srucure was series. A ne, consider he (parallel) sysem s residual life -ime s failure rae, say r k () i.e., he failure rae of eiher surviving componen c k, a any ime saisfying y, ( where, y are he ime epochs of he firs and he second failure in he sysem respecively ). For ha period of he ime we have chosen he following failure paern. Namely, we define he failure rae r k () in he ime inerval [, y] as he following arihmeic sum: r k () = λ k ( ) + 0 α k m ( τ ) d τ. (5) In his cone, he inegral 0 α k m ( τ ) d τ is consan over ime pas ( k, m =,, and k m ). The reason for is consancy is based on a simple observaion ha in he ime inerval [, y] only (one) componen c k is working in he sysem, and hus he process of micro -damages accumulaion is erminaed since he ime passed. This inegral (presen as a par in (6), (6a) ha follow) is an addiional par of he overall failure rae r k () of componen c k, and may be undersood as a measure of memory of he microincenives he c k received before he oher componen c m sopped is aciviy a ime. The Final Formula for he join survival funcion S(, ) = Pr(X >, X > ) of he in-sysem componen life-imes X, X, is given below as: Pr (X >, X > ) = ep [ - 0 { λ ( ) + 0 α, (τ ) d τ }d - 0 { λ ( ) + 0 α, (τ ) d τ } d ] ep [ - { λ ( ) d } (6) - ( - ) 0 α, (τ ) d τ ] ; when, and Pr(X >, X > ) = ep [ - 0 { λ ( ) + 0 α, (τ ) d τ } d - 0 { λ ( ) + 0 α, (τ ) d τ } d ] ep [ - { λ ( ) d } - ( - ) 0 α, (τ ) d τ ], when >. (6a) If in boh formulas (6) and (6a) one ses = 0, hen one obains he marginal probabiliy disribuion of X o be he same as he original probabiliy disribuion F ( ) of he off-sysem life-ime T, relaed o he given in advance original failure rae λ ( ). The similar resul one obains when imposing in (6), (6a) he condiion = 0.

5 The laer condiion yields he marginal disribuion of he X o be equal F ( ). As he conclusion one derives he following surprising Propery, shared by all he models ha obey he paern epressed by (6), (6a). Propery. For any join probabiliy disribuion given by he join reliabiliy funcion S(, ) defined by (6), (6a), any, given in advance, original probabiliy disribuions F ( ), F ( ) of he offsysem life-imes T, T (wih he assumed noaion: =, = ) are preserved! as he marginal disribuions of he join probabiliy disribuion of he in-sysem life-imes X, X of he considered unis u, u. From Propery, he conclusion can be derived as he following: Corollary. Suppose we are given a pair of probabiliy disribuions G ( ), G ( ) ha belong o any class of probabiliy disribuion funcions, whose all members posses coninuous failure (hazard) raes, say λ ( ), λ ( ). If one pus any arbirary single pair of such disribuions ino he scheme defined by (7), (7a) hen, as a resul, one can generae a wide class of he bivariae survival funcions S(, ), whose marginals remain o be he G ( ), G ( ). The class of he so obained bivariae probabiliy disribuions, given he (fied) marginals G ( ), G ( ), is deermined by he family of all he coninuous funcions α i, j (τ i ), ( i, j =,, wih i j ) ha produce all he inegrals in (7), (7a) finie. So, in his paricular sense one can consider he bivariae Weibull, gamma (in paricular, eponenial), he ereme value and oher join probabiliy disribuions. Realize, however ha he marginal disribuions G ( ), G ( ), in he Corollary also may represen wo disinc disribuion classes. The las possibiliy may be uilized in modeling reliabiliy of wo sochasically dependen unis (such as sysem componens) each one subjeced o a differen failure mechanism. Apparenly such cases ofen are realisic. III. EXAMPLE As a paricular class of he bivariae survival funcions S(, ), saisfying he paern given by (6), (6a), we now choose a class of bivariae eponenial disribuions, given by wo arbirary consan failure raes λ, λ for he marginals. We also resric he dependence srucure, by assuming i is only deermined by wo consan funcions α, ( ), α, ( ). Recall, hey represen he raes of incremen in he failure rae of he uni u caused by u and he failure rae incremen of u, caused by u, respecively. The resuling class of he join eponenial survival funcions can be epressed by he following specificaion of he previous, more general, paerns (6), (6a) : Pr(X >, X > ) = ep [ - 0 { λ + 0 α, d τ } d - 0 { λ + 0 α, d τ } d ] (7) ep [ - { λ d } - (α, ) ( - ) ] for, and

6 Pr(X >, X > ) = ep [ - 0 { λ + 0 α, d τ } d - 0 { λ + 0 α, d τ } d ] (7a) ep [ - { λ d } - (α, ) ( - )], for >. Upon addiionally simplifying he assumpion saing ha α, ( ) = α, ( ) = α = consan, boh formulas (7) and (7a) reduce o he following single one: Pr(X >, X > ) = ep [ - λ - λ - α ] (8) Thus, as a special case of he model one obains he firs bivariae eponenial Gumbel probabiliy disribuion as sysem reliabiliy model. IV. REMARK There is an ineresing relaionship beween he models defined in his paper (in paricular he Gumbel model) and he model given by Freund in [5]. In Freund model he wo componens work independenly (no ineracions) unil he firs failure and hen he remaining componen is affeced by lack of he oher componen. Unlike in Freund s case we have eacly opposie, say complemenary failure mechanism. The componens inerac as long as hey work ogeher while afer he firs failure, he remaining componen enjoys he off sysem (normal) siuaion. Also, he consrucions presened here differ significanly from he models considered in [] or in similar papers cied in []. REFERENCES [] B. C. Arnold, E. Casillo, and J. M. Sarabia, Condiionally Specified Disribuions, Lecure Noes in Saisics - 73, New York: Springer- Verlag, 99. [] R. E. Barlow and F. Proschan, Saisical Theory of Reliabiliy and Life Tesing, Hol, Rinehar and Winson, New York, 975. [3] J. K. Filus, On a Type of Dependencies beween Weibull Life imes of Sysem Componens, Reliabiliy Engineering and Sysem Safey, Vol.3, No.3, 99, pp [4] J. K Filus, L. Z. Filus, On Some New Classes of Mulivariae Probabiliy Disribuions. Pakisan Journal of Saisics,, Vol., 006, pp. 4. [5] J. E. Freund, A Bivariae Eension of he Eponenial Disribuion, J. Amer. Sais. Assoc., Vol 56, 96, pp [6] G. Heinrich and U. Jensen, Parameer esimaion for a bivariae lifedisribuion in reliabiliy wih mulivariae eensions, Merika, Vol 4, 995, pp [7] A. W. Marshall and I. Olkin, A Generalized Bivariae Eponenial Disribuion, Journal of Applied Probabiliy 4, 967, pp